Spherical bessel functunction help

Then use the same recurrence relation to eliminate J_{n+1/2}(x) in favor of J_{n-1/2}(x) and J_{n+3/2}(x). Expanding these terms and simplifying should result in the desired recurrence relation. In summary, the conversation discusses the process of deriving recurrence relations for the spherical Bessel function. The person is struggling to understand where the 'n's come from in the recurrence relation and is seeking guidance on where to start. The suggested approach is to compute j_n'(x) in terms of J_{n+1/2}(x) and then use the ordinary recurrence relation to eliminate J'_{n+1/2}(x) and
  • #1
lycraa
17
0

Homework Statement


i need to derive the recurrence relations for the spherical Bessel function. i got jn-1(x)+jn+1(x)=(2n+1)/x jn(x) but i can't get njn-1(x)-(n+1)jn+1(x)=(2n+1) j'n(x). i know i have to use jn(x)=(pi/2x)1/2Jn+1/2(x) and the recurrence relations for regular bessel functions Jn-1(x)-Jn+1(x)=2 J'n(x) and possibly also Jn-1(x)+Jn+1(x)=2n/x Jn(x). i don't even know were to start because i don't know were the 'n's come from in the recurrence relation I'm trying to derive.


Homework Equations


above

The Attempt at a Solution


i just need to know were to start! nothing I've tried comes even close
 
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  • #2
Start by computing [tex]j_n'(x)[/tex] in terms of [tex]J_{n+1/2}(x)[/tex]. Use the ordinary recurrence relation to eliminate [tex]J'_{n+1/2}(x)[/tex].
 

1. What is a spherical bessel function?

A spherical bessel function is a mathematical function that describes the behavior of spherical waves. It is commonly used in physics and engineering to solve problems involving spherical symmetry, such as in electromagnetic or acoustic wave propagation.

2. How is a spherical bessel function different from a regular bessel function?

A spherical bessel function is a special case of the regular bessel function, where the argument of the function is multiplied by the radius of a sphere. This takes into account the three-dimensional nature of the problem, whereas a regular bessel function only considers the one-dimensional case.

3. What are the applications of spherical bessel functions?

Spherical bessel functions are commonly used in fields such as acoustics, electromagnetics, quantum mechanics, and fluid dynamics. They are particularly useful in solving problems with spherical symmetry, such as in the analysis of spherical antennas, sound radiation from a point source, and the scattering of electromagnetic waves by a spherical object.

4. How do I calculate a spherical bessel function?

There are several ways to calculate a spherical bessel function, depending on the specific problem at hand. One way is to use a computer software program, such as MATLAB or Mathematica, which have built-in functions for calculating spherical bessel functions. Another way is to use a table of pre-calculated values or a specialized calculator. Alternatively, you can use a series expansion or recurrence relation to approximate the function.

5. Are there any special properties of spherical bessel functions?

Yes, there are several special properties of spherical bessel functions that make them particularly useful in solving problems. For example, they are orthogonal to each other, meaning that the integral of their product over a specific range is equal to zero. They also have a specific symmetry that allows for simplification in certain calculations. Additionally, they have a singularity at the origin, which can be problematic in some cases but can also provide valuable information about the behavior of a system.

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